Question

suppose that you have 7 green cards and 5 yellow cards. the cards are well shuffled. you randomly draw two cards without replacement. round answers to 4 decimal places.
g1 = the first card drawn is green
g2 = the second card drawn is green
a. p(g1 and g2) =

b. p(at least 1 green) =

c. p(g2|g1) =

d. are g1 and g2 independent?

they are independent events
they are dependent events

hint: independent events
video on independent events -

Explanation:

Step1: Calculate total number of cards

Total cards = 7 (green) + 5 (yellow)=12 cards.

Step2: Calculate P(G1 and G2)

P(G1)=$\frac{7}{12}$, after drawing a green - card first, there are 6 green cards left and 11 total cards left. So P(G2|G1)=$\frac{6}{11}$. By the multiplication rule P(G1 and G2)=P(G1)×P(G2|G1)=$\frac{7}{12}\times\frac{6}{11}=\frac{42}{132}\approx0.3182$.

Step3: Calculate P(At least 1 green)

The complement of "at least 1 green" is "no green (both yellow)". P(both yellow)=$\frac{5}{12}\times\frac{4}{11}=\frac{20}{132}$. So P(At least 1 green)=1 - P(both yellow)=1 - $\frac{20}{132}=\frac{112}{132}\approx0.8485$.

Step4: Calculate P(G2|G1)

As calculated in Step 2, P(G2|G1)=$\frac{6}{11}\approx0.5455$.

Step5: Determine independence

Two events G1 and G2 are independent if P(G1 and G2)=P(G1)×P(G2). P(G1)=$\frac{7}{12}$, P(G2)=$\frac{7}{12}$ (if events were independent). But P(G1 and G2)=$\frac{7}{12}\times\frac{6}{11}
eq\frac{7}{12}\times\frac{7}{12}$, so they are dependent events.

Answer:

a. 0.3182
b. 0.8485
c. 0.5455
d. They are dependent events